A Linear Algorithm for Tracing Magnet Position and Orientation by Using Three-Axis Magnetic Sensors

4096IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 12, DECEMBER 2007A Linear Algorithm for Tracing Magnet Position and Orientation by Using Three-Axis Magnetic SensorsChao Hu1;3 , Max Q.-H. Meng2;3 , and Mrinal Mandal1Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada Department of Electronic Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong Institute of Advanced Integration Technology/Shenzhen Institute of Advanced Technology, Shenzhen 518067, ChinaFor medical diagnoses and treatments, it is often desirable to wirelessly trace an object that moves inside the human body. A magnetic tracing technique suggested for such applications uses a small magnet as the excitation source, which does not require the power supply and connection wire. It provides good tracing accuracy and can be easily implemented. As the magnet moves, it establishes around the human body a static magnetic field, whose intensity is related to the magnet’s 3-D position and 2-D orientation parameters. With magnetic sensors, these magnetic intensities can be detected in some predetermined spatial points, and the position and orientation parameters can be computed. Typically, a nonlinear optimization algorithm is applied to such a problem, but a linear algorithm is preferable for faster, more reliable computation, and lower complexity. In this paper, we propose a linear algorithm to determine the 5-D magnet’s position and orientation parameters. With the data from five (or more) three-axis magnetic sensors, this algorithm results in a solution by the matrix and algebra computations. We applied this linear algorithm on the real localization system, and the results of simulations and real experiments show that satisfactory tracing accuracy can be achieved by using a sensor array with enough three-axis magnetic sensors. Index Terms—Linear algorithm, magnet’s position and orientation, magnetic tracing.I. INTRODUCTIONRECENTLY, many researchers have suggested the magnetic tracing (MT) technique in medical applications [1]–[8]. This is because the human body has a magnetic permeability very close to that of air, water, or nonferromagnetic material, and exerts very little influence on the static (or low-frequency) magnetic field, such that it is possible to achieve high tracing accuracy. Nevertheless, the MT technique is of higher speed and easier realization compared to other possible techniques, e.g., computer tomography, magnetic resonance imaging, and 3-D ultrasonic techniques. Typically, this technique uses magnetic excitation sources, which can be created by one single magnetic dipole (or coil), or two or three orthogonal magnetic dipoles (or coils). The magnetic field signals produced by these dipoles can be measured by the magnetic sensors (or receiving coils) arranged in some positions around the human body. With the measured sensor data, the tracing parameters can be computed using an appropriate algorithm. In some specific applications, it is difficult to use cabled equipment, making the wireless technique preferable, e.g., the tracking of wireless capsule endoscopes [9], and the monitoring of the pill transit [10]–[15]. Since such applications demand as low a power and space as possible, a permanent magnet is therefore a reasonable choice to serve as the excitation dipole [9], [11], [13]–[15] instead of the electric current coil [16]–[21]. The magnetic field created by a permanent magnet is a highorder nonlinear function of the 5-D magnet’s position and orientation parameters. To solve such a problem, the nonlinear least square optimization algorithm is a common method [10]–[15].However, the nonlinear optimization method has its limitations. First, it must have an initial estimation of the parameters (or their bounds), to begin the search for the optimum. If the initial parameters have a large error, the algorithms might fail to give a correct global solution because the mathematic model of the magnetic dipole has many local minima. Second, the searching process takes a long time, meaning that there is not enough time to apply other signal processing measures to improve the tracing accuracy. Third, the computation complexity might be so high that it becomes difficult to be realized in a real-time instrument. All these problems can be solved by a linear algorithm. Therefore, in this paper, we propose a linear algorithm for the magnet’s localization and orientation using five or more three-axis magnetic sensors. The proposed algorithm is derived by the absolute vector computation, so no calculation error is induced by the approximations in the equation derivation. The organization of this paper is as follows. In Section II, we introduce the mathematical model of the magnetic field produced by a permanent magnet. In Section III, we propose the linear algorithm. In Section IV, we present the performance of the algorithm by an example and the results of simulation and real experimentation, which is followed by the conclusions in Section V. II. NONLINEAR MATHEMATIC MODEL Fig. 1 shows a cylindrical magnet with length , diameter , and a uniform magnetization (Amp/meter) on the top and bottom surfaces. Let be the vector that represents a spatial point with respect to the magnet center position . The magnetic flux density at this point can be calculated as [22] (1)Digital Object Identifier 10.1109/TMAG.2007.907581 Color versions of one or more of the figures in this paper are available online at .where ability of the medium,is the relative permeis the magnetic permeability0018-9464/$25.00 © 2007 IEEEHU et al.: A LINEAR ALGORITHM FOR TRACING MAGNET POSITION AND ORIENTATION4097is orthogonal to , making the dot product Because , we get on both sides of the above equation with (8) That isFig. 1. Coordinate system for magnet’s localization: (a; b; c) is the position of is the direction of the magnet. p = (x ; y ; z ) represents a the magnet, spatial point.H(9) of the air, and is the normalized vector representing the orientation of the magnet’s magnetism. Assume that there are magnetic sensors, with th sensor located at . The magnetic flux density at the th sensor location can be represented by (2) , and are the magnetic components in the where three orthogonal coordinate axes, and could be represented by We observe that in this equation, the magnitudes of these vecand are irrespective to the solution. However, tors it gives some relative relations of the parameters of these vectors. This means that we cannot get the unique true solution only using (8). To get the unique solution, we need to use other constraints (these constraints are defined by (6), (28), (29), and will be further presented below). Expanding (9), we have(10) (3) It can be further simplified as (11) where (5) (12) . where Note that there are six unknown parameters . The flux intensity is invariant to the rotation of the magnet along its major axis. Hence, the magnet’s orientation is in two dimensions, in other words, the length of vector can be any fixed value. Therefore, we add the following constraint to : (6) , and Since the magnetic intensity components can be measured by the magnetic sensors, we can apply a nonlinear algorithm to solve for the six localization and orientation parameters using these equations. The most common approach is to define an objective error function by changing (3)–(6) into the form of summation of square errors, and applying an optimization algorithm to find the parameters that minimize the error function (the least square error method) [9]. However, as mentioned in Section I, the nonlinear algorithm has its limitations, so instead we propose a linear algorithm. III. LINEAR ALGORITHM A. Derivation of the Linear Model Because sides of (1) with , making the cross product on both , we have (7) (13) (14) . In (11), and are funcwith in the tions of the magnetic components , and independent of the six unspatial position is only composed of known parameters the six unknown parameters. In some predetermined posi, we can dispose three-axis tions magnetic sensors to measure the magnetic components , and , such that and can be determined. Therefore, with five three-axis magnetic sensors, a matrix and a vector are formed, and then we solve for (15) We have tested the solution with the simulations using different conditions, and found that the unique solution can be obis nonsingular. In cases in which is tained if the matrix singular, we take the following measures. , and , then we 1) We define , and : have different representations of (16) (17) (18)(4)4098IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 12, DECEMBER 2007or we define then, and, and (19) (20) (21)The matrix is singular (we can verify ), but its rank is two, so we can get two independent equations (26) (27) To solve for parameter , we need to find another equation. Dividing (3) and (4) with (5), we have the following two equations:2) Weusesix,ormorethansix,three-axismagneticsensors.Becausethesensornumberismorethanthenumberofunknown parameters,thesingularitycanbeavoidedifthesesensorsare arranged in specifically-selected spatial positions. Once is solved, the magnet’s position and orientation parameters , and can further be solved. This method is much simpler than solving nonlinear equations (3)–(6) directly. The solution is also irrelevant to the magnetization of the magnet and the relative permeability of the medium. The cost of this method is that it requires five three-axis sensors as a minimum, while the nonlinear method could use five sensors that could be one-axis, or two-axis or three-axis sensors. B. Algorithm for Magnetic Sensor Array Although five three-axis magnetic sensors are enough for solving (11), more sensors are preferable to get higher tracking accuracy, because the accuracy is influenced by the interferences in the sensor signals, especially in the case of low signal-to-noise ratio (SNR). From (1), we observe that magnetic intensity is an inverse function with the power of three of the distance from the magnet to the sensor. In order to obtain stronger sensor signals for high SNR, we should control this distance with more densely distributed sensors. To simplify the problem, these sensors can be arranged in a plane so as to form a sensor array [9], [13], [16]. When there are more than five sensors, we can apply the following optimal fitting method. Assume there are totally three-axis sensors, we change the expression of (11), and define the fitting error to be . The optimal solution of which makes minimum is (22) is 5 where element is with and th element is given by (14). 5 matrix, in which the th row and th column given by (12); is a vector, in which the with(28) and(29) Let (26) and (27) be substituted into (28) and (29), giving us the following 2-order equations with one unknown : (30) (31) are the functions of known as well as the solutions of . We can select any one, or two equations to solve , among the 2 equations defined by (30) or (31). Here, we present a more accurate method to determine by applying the least square error method. We define an error to be (32) Using this equation, three roots of can be solved for a minimum of . If there are two imaginary roots and one real root (in most cases), the solution is the real root; if all three roots are real, they are tried in (32), and the solution is chosen as the one with the minimum of . With this , we can compute and by (26) and (27). D. Determine the Sign of the Orientation From (23), we know that the calculated orientation parameter (and ) could be positive or negative. This is not true in real cases, where the magnet has its unique magnetic direction from one surface to another surface. Hence, we need to determine the sign of the orientation. Here, we first show that the sign of the does not affect the calculation of orientation . the location parameters and , (28) and (29) can be Using rewritten as where data andC. Solution of the Magnet Position and Orientation Now we have the solutions of (6), we get the orientation . From (23) (24) (25) Here, the orientation might have a positive or negative sign. We will discuss how to determine this sign in Step D below. , from (10), we have For the magnet position(33)(34)HU et al.: A LINEAR ALGORITHM FOR TRACING MAGNET POSITION AND ORIENTATION4099Because are calculated using these two equations and , the sign of does not the solution of . affect the calculation results of and Now, with the calculated results of , we define and calculateUsing the sensor positions and the above sensor outputs, , which is singular because we get the matrix and and are infinite. Here, we adopt (16)–(18), and the solutions of is . Then the orientation is calculated as(35)(36)(37) From (3)–(5), we know (38) (39) (40) Then, it is clear thatwhere the sign of orientation is wrong as compared to given data, so we will adjust it later. With this orientation, we calculate the parameter by the least square error method [in (32)], , and and reach three roots: . The solution is the real root, we obtain the parameter . Then, two other position parameters can be solved as , and . With these parameters, we findsuch thatis negative. That is(41) Therefore, the sign of the orientation can be determined. If ; else . When the sign of is determined, the signs and can also be adjusted. Therefore, the six unknowns are solved. IV. PERFORMANCE EVALUATION BY SIMULATION AND EXPERIMENTS In this section, we evaluate the performance of the proposed algorithm by giving an example and the results of simulation and real experiments. A. Example Here, an example is presented for five three-axis sensors to be applied to the localization and orientation. The five sensors are located in and (in meters). The magnet’s position and orientation is given as . From (3)–(5), we get 5-sensor outputs (relative to ) asHence, the resultant localization and orientation parameters are . This is exactly the same as the pre-setup parameters. B. Simulation for Denoising Ability The algorithm was tested in MATLAB programs, which uses magnetic intensities created by some magnets’ positions and directions. To test the anti-interference ability of the algorithm, white-noise was added in the simulated sensor signals. The algorithm is tested using 5, 9, and 16 sensors. For the 5-sensors’ case, the sensors are arranged on the center and 0.2 m. In four corners of a square plane with an area of 0.2 m the case of 9-sensors and 16-sensors, the sensors are uniformly arranged on the square plane of 0.2 m 0.2 m and 0.24 m 0.24 m, as shown in Figs. 2 and 3. To evaluate the performances, we define the localization error and orientation error to be (42) (43) Figs. 4–6 show simulation results of the localization error via the noise level in the magnetic intensity. Here, we represent the noise relative to the magnet’s constant . In the 5-sensor case, the localization (position) errors are too large ( mm) if the noise level is larger than 18. This error is much improved when more sensors are used. As shown in Figs. 5 and 6, when the noise level is within 65 times of Bt, the localization error is smaller than 18 mm in the 9-sensors case, and 3 mm in the 16-sensors case.of4100IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 12, DECEMBER 2007Fig. 2. Arrangement of 9-sensors.Fig. 6. Localization accuracy by 16-sensors.Fig. 3. Arrangement of 16-sensors. Fig. 7. 3-D localization results.Fig. 4. Localization accuracy by 5-sensors.Fig. 8. 2-D (X-Y plane) localization results.Fig. 5. Localization accuracy by 9-sensors.C. Real Experiment We built an experimental system, which includes magnetic (Hall) sensors, a precision amplification circuit, a stabilized power source, AD converters, and a PC computer. The sixteen three-axis sensors are fixed in a plane such that a 2-D array is formed as illustrated in Fig. 3. We get sensing data from these sensors when the magnet is located in some spatial positions and orientations that can be determined. These data are filteredusing moving average techniques so that they produce much less noise. Figs. 7 and 8 show the 3-D and 2-D localization results. Figs. 9 and 10 show the localization and orientation errors at different positions. We observe that within the sensor area, the localization error is within 8 mm, and the orientation error is within 7%. The average localization error is 5.6 mm and the average orientation error is 2.9% (around 1.7 ). While comparing to the nonlinear algorithm, similar accuracy (5.6 mm, 4.26% in nonlinear algorithm [9]) is obtained for cases in which the nonlinear algorithm adopts a proper initial guess of the parameters. However, the execution speed of the proposed linear algorithm is much faster. Through the simulation, we observed that the execution time is only one tenth of that of the fastest nonlinear algorithm, so that the linear algorithm can provide greater capacity for signal processing to obtain further improved accuracy.HU et al.: A LINEAR ALGORITHM FOR TRACING MAGNET POSITION AND ORIENTATION4101Fig. 9. Localization error.Fig. 10. Orientation error.V. CONCLUSION A linear algorithm is proposed to calculate a magnet’s position and orientation. This algorithm is realized by simple matrix and algebraic calculations, and can be applied to the magnetic sensor array composed of five or more three-axis magnetic sensors. Higher accuracy can be obtained with denser sensor arrangements. In addition, a real test system has been built by an array of sixteen three-axis Hall sensors, and results are satisfactory. Compared to the nonlinear algorithms, this algorithm has faster speed, lower complexity, and higher reliability. Using this algorithm, we are now building a real time system for tracking a capsule that moves in the human GI tract. ACKNOWLEDGMENT This project was supported by RGC Competitive Earmarked Research Grant #CUHK4213/04E of the Hong Kong government, awarded to Max Meng. REFERENCES[1] NDI Aurora Electromagnetic Tracking System. [Online]. Available: /aurora.php [2] Ascension Technol. Corp. Products Application. [Online]. Available: /products/microbird.php [3] D. D. Frantz, A. D. Wiles, S. E. Leis, and S. R. Kirsch, “Accuracy assessment protocols for electromagnetic tracking systems,” Phys. Med. Biol., vol. 48, pp. 2241–2251, 2003.[4] K. Schicho, M. Figl, M. Donat, W. Birkfellner, R. Seemann, A. Wagner, H. Bergmann, and R. Ewers, “Stability of miniature electromagnetic tracking systems,” Phys. Med. Biol., vol. 50, pp. 2089–2098, 2005. [5] J. Hummel, M. Figl, C. Kollmann, and H. Bergmann, “Evaluation of a miniature electro-magnetic position tracker,” Med. Phys., vol. 29, no. 10, pp. 2205–2212, 2002. [6] K. H. Wong, J. Tang, S. Dieterich, H. Zhang, T. Zhou, and K. Cleary, “Respiratory motion compensation studies using a 3D robotic motion simulator and optical/electromagnetic tracking technologies,” in 2004 IEEE Nucl. Sci. Symp. Conf., Rome, Italy, Oct. 22–26, 2004. [7] D. C. Barratt, A. H. Davies, A. D. Hughes, S. A. Thom, and K. N. Humphies, “Optimisation and evaluation of an electromagnetic tracking device for high-accuracy three-dimensional ultrasound imaging of the carotid arteries,” Ultrasound Med. Biol., vol. 27, no. 7, pp. 957–968, Jul. 2001. [8] Y. Yamada, N. Yoshida, K. Kobayashi, and K. Yamauchi, “An application of magnet and magnetic sensor: Measurement system for tooth movement,” IEEE Trans. Biomed. Eng., vol. 37, no. 10, pp. 919–924, Oct. 1990. [9] C. Hu, M. Q.- H. Meng, and M. Mandal, “Efficient magnetic localization and orientation technique for capsule endoscopy,” Int. J. Inf. Acquisition, vol. 2, no. 1, pp. 23–36, 2005. [10] W. Weitschies, M. Karaus, D. Cordini, L. Trahms, J. Breitkreutz, and W. Semmler, “Magnetic marker monitoring of disintegrating capsules,” Eur. J. Pharm. Sci., vol. 13, no. 4, pp. 411–416, Jul. 2001. [11] W. Weitschiles, R. Kotitz, DinoCordini, and L. Trahms, “High-resolution monitoring of the gastro-intestinal transit of a magnetic marked capsule,” J. Pharm. Sci., vol. 86, no. 1, pp. 1218–1222, 1997. [12] W. Weitschies, J. Wedmeyer, R. Stehr, and L. Trahms, “Magnetic markers as a noninvasive tool to monitor gastrointestinal transit,” IEEE Trans. Biomed. Eng., vol. 41, no. 2, pp. 192–195, Feb. 1994. [13] V. Schlageter, P.-A. Besse, R. S. Popovic, and P. Kucera, “Tracking system with five degrees of freedom using 2D-array of Hall sensors and a permanent magnet,” Sens. Actuators, vol. 92, pp. 37–42, 2001. [14] N. Mani Prakash and F. A. Spelman, “Localization of a magnetic marker for GI motility studies: An in vitro feasibility study,” in Proc. 19th Annu. Int. Conf. IEEE Eng. Medicine Biol. Soc., Chicago, IL, Oct. 30–Nov. 2, 1997, pp. 2394–3239. [15] W. Andra, H. Danan, W. Kirmße, H.-H. Kramer, P. Saupe, R. Schmieg, and M. E. Bellemann, “A novel method for real-time magnetic marker monitoring in the gastrointestinal tract,” Phys. Med. Biol., vol. 45, pp. 3081–3093, 2000. [16] A. Plotkin and E. Paperno, “3D magnetic tracking of a single subminiature coil with a large 2-D array of uniaxial transmitters,” IEEE Trans. Magn., vol. 39, no. 5, pp. 3295–3297, Sep. 2003. [17] J. E. McFee and Y. Das, “Determination of the parameters of a dipole by measurement of its magnetic field,” IEEE Trans. Antennas Propag., vol. AP-29, no. 2, pp. 282–287, Mar. 1981. [18] Y. Liu, Y. Wang, D. Zhou, X. Hu, and J. Wu, “Study on an experimental AC electromagnetic tracking system,” in Proc. 5th World Congr. Intelligent Control Autom., Hangzhou, China, Jun. 15–19, 2004. [19] F. H. Raab, E. B. Blood, T. O. Steiner, and H. R. Jones, “Magnetic position and orientation tracking system,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-15, no. 5, pp. 709–718, Sep. 1979. [20] E. Paperno, I. Sasada, and E. Leonovich, “A new method for magnetic position and orientation tracking,” IEEE Trans. Magn., vol. 37, no. 4, pp. 1938–1940, Jul. 2001. [21] H. P. Kalmus, “A new guiding and tracking system,” IRE Trans. Aerosp. Navig. Electron., vol. 9, pp. 7–10, Mar. 1962. [22] D. K. Cheng, Field and Wave Electromagnetics: Static Magnetic Fields. Reading, MA: Addison-Wesley, 1989.Manuscript received November 23, 2005; revised August 28, 2007. Corresponding author: C. Hu (e-mail: hcsfds@ece.ualberta.ca; hu.chao.hu@gmail. com).。